Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Notation 2.5.6.11. Let $M_{\ast }$ be a chain complex. Then every $n$-simplex $\sigma $ of the simplicial set $\mathrm{K}( M_{\ast } )$ can be identified with a map of chain complexes $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$, which carries the generator of $\mathrm{N}_{n}( \Delta ^ n; \operatorname{\mathbf{Z}})$ to an $n$-chain $\widetilde{v}(\sigma ) \in M_ n$. Moreover:

  • Since $\sigma $ is a morphism of chain complexes, we have

    \[ \partial ( \widetilde{v}(\sigma ) ) = \sum _{i =0}^{n} (-1)^{i} \widetilde{v}( d_ i \sigma ). \]

    In other words, the construction $\sigma \mapsto \widetilde{v}(\sigma )$ determines a chain map from the Moore complex $\mathrm{C}_{\ast }( \mathrm{K}( M_{\ast } ) )$ to the chain complex $M_{\ast }$.

  • If $\sigma $ is a degenerate $n$-simplex of $\mathrm{K}( M_{\ast } )$, then the map of chain complexes $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow M_{\ast }$ factors through $\mathrm{N}_{\ast }( \Delta ^{m}; \operatorname{\mathbf{Z}})$ for some $m < n$, and therefore annihilates the generator of $\mathrm{N}_{n}( \Delta ^ n; \operatorname{\mathbf{Z}})$. It follows that $\widetilde{v}$ factors (uniquely) as a composition

    \[ \mathrm{C}_{\ast }( \mathrm{K}( M_{\ast } ) ) \twoheadrightarrow \mathrm{N}_{\ast }( \mathrm{K}(M_{\ast }) ) \xrightarrow {v} M_{\ast }. \]

We will refer to the chain map $v: \mathrm{N}_{\ast }( \mathrm{K}(M_{\ast }) ) \rightarrow M_{\ast }$ as the counit map.